I.  , 


*^ 


i 


c 

'/ 


^XCHANGIS 
MAR  8     J917 

COMPLEX   CONICS   AND   THEIR 


REAL  REPRESENTATION 


BY 


BENJAMIN   ERNEST   MITCHELL     ^<.- 


DISSERTATION 

Submitted  in  Partial  Fulfillment  of  the  Requirements 

FOR  THE  Degree  of  Doctor  of  Philosophy, 

in  the  Faculty  of  Pure  Science, 

Columbia  University 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY 

UAJMCASTER,  PA. 


1917 


1) 


COMPLEX   CONICS   AND   THEIR 
REAL   REPRESENTATION 


BY 

BENJAMIN    ERNEST   MITCHELL 


DISSERTATION 

Submitted  in  Partial  Fulfillment  of  the  Requirements 

FOR  THE  Degree  of  Doctor  of  Philosophy, 

IN  THE  Faculty  of  Pure  Science, 

Columbia  University 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY 

LANCASTER,  PA. 

1917 


TABLE  OF  CONTENTS. 

Chapter 

Introduction 1 

1.  Historical 1 

2.  The  Laguerre-Study  Representation  of  the  Imag- 

inary quantity 2 

I.     The  Real  Conies 6 

3.  Some  Simple  Cases 6 

4.  The  Real  Ellipse 7 

5.  The  Real  Hyperbola 13 

6.  The  Real  Parabola 15 

II.    The  Complex  Conic,  Preliminary  Considerations 18 

7.  The  Complex  Line  and  Its  Conjugate 18 

8.  Canonical  Equation  of  the  Complex  Line 19 

9.  Real  Representation  of  the  Complex  Line 21 

10.  Canonical  Equation  of  the  Complex  Circle 21 

11.  The  Complex  Circle  and  Its  Conjugate 22 

12.  Reflection  with  Respect  to  the  Complex  Circle. .  23 
III.     The  Complex  Conic,  Reduction  of  Equation 27 

13.  The  Complex  Conic  and  Its  Conjugate 27 

14.  Canonical  Equation  of  the  Complex  Conic 29 

15.  Localizing  the  Complex  Conic 32 

IV.    The  Complex  Conic,  Its  Real  Representation 33 

16.  The  General  Case,  Z  4=  0 33 

17.  The  Special  Case,  Z  =  0 42 

18.  The  Special  Case,  6  =  0 44 


o4:.>i^'J 


COMPLEX  CONICS  AND  THEIR  REAL 
REPRESENTATION. 


INTRODUCTION. 

1.  Historical. — The  introduction  of  the  imaginary  quantity, 
or  the  complex  quantity  comprehending  both  the  real  and  the 
imaginary,  into  analysis  had  the  effect  not  only  of  extension 
and  generalization  but  also  in  many  cases  of  simplification. 
Such  results  in  the  realm  of  geometry  have  not  yet  been  fully 
realized.  In  the  preface  to  his  "Einfiihrung  in  die  analytische 
Geometric,"  Kowalewski  says:  "Eine  grosse  Schwierigkeit  in 
der  analytischen  Geometric  ist  die  exakte  Behandlung  des 
Imaginaren." 

But  the  incorporation  of  the  imaginary  in  geometry  does  not 
require  any  more  of  reconstruction  and  readjustment  than  it  did 
in  the  case  of  analysis.^  "A  satisfactory  theory  of  imaginary 
quantities  of  the  ordinary  algebra  .  .  .  with  difficulty  obtained 
recognition  in  the  first  third  of  this  century  .  .  .  ,  it  .  .  .  was 
not  sought  for  or  invented — it  forced  itself,  unbidden,  upon  the 
attention  of  mathematicians,  and  with  its  rules  already  formed."'" 

No  sooner  had  the  imaginary  won  its  rightful  place  in  analysis 
at  the  hand  of  Gauss  and  Cauchy  than  it  began  to  knock  at  the 
door  of  geometry.  Indeed  before  its  full  recognition  in  analysis 
there  had  appeared  the  geometric  method  of  representing  the 
imaginary  quantity  due  to  Argand  and  Wessel.  But  the  imag- 
inary in  geometry  must  play  the  role  of  element  of  structure 
comparable  to  that  of  number  in  analysis.^  The  history  of  its 
development  is  of  intense  interest. 

1  Convention.  Toute  expression  ayant  un  sens  geometrique  quand  les 
elements  dont  elle  depend  sont  reels  conservera,  par  definition,  le  meme  nom 
quand  quelques-uns  de  ces  elements  deviendront  imaginaries.  Niewenglowski , 
Cours  de  Geometric  Analytique,  p.  114,  Old  Edition. 

2  Gibbs:  "On  Multiple  Algebra,"  Proc.  Am.  Asso.  Adv.  of  Sci.,  1886. 

3  Cf.  C.  A.  Scott:  "On  Von  Staudt's  Geometric  der  Lage,"  Math.  Gazette, 
Vol.  1,  p.  307. 

1 


•.^.'.-"  COMRLEX  CGNICS  AND  THEIR  REAL  REPRESENTATION. 

Out  of  the  school  of  Monge  came  a  class  of  mathematicians, 
beginning  with  Poncelet  and  culminatmg  in  Von  Staudt,  who 
were  somewhat  exclusive  in  their  methods.  On  the  principle, 
geometry  for  geometers  and  geometry  all-sufficient  and  self- 
sufficient,  they  set  about  to  build  up  a  body  of  doctrine  wholly 
independent  of  analysis.  The  principals  in  this  program  were 
Poncelet,  Chasles,  Steiner  and  Von  Staudt.  The  imaginary 
enters  through  the  so-called  Principle  of  Continuity  and  makes 
its  first  appearance  in  Chasles'  "Traite  de  Geometric  Superieure" 
(1852).  Regarding  the  second  of  the  three  advantages  which 
he  claims  for  his  geometry  he  says:  "  Je  veux  parler  de  la  gener- 
alite  dont  sont  empreints  tons  les  resultats  de  la  geometric 
analytique,  oii  Ton  ne  fait  acception  ni  des  differences  de  positions 
relatives  des  diverses  parties  d'une  figure,  ni  des  circonstances  de 
realite  ou  d'imaginarite  des  parties,  qui,  dans  la  construction 
generale  de  la  figure,  peuvent  etre  indifferemment  reelles  ou 
imaginaries.  Ce  caractere  specifique  de  I'Analyse  se  trouve  dans 
notre  Geometric." 

But  the  investigations  of  Poncelet  and  Chasles  had  their 
origins  in  analysis  and  in  their  completed  forms  were  not  free 
from  analytical  considerations.  To  George  Karl  Christain  Von 
Staudt  belongs  the  honor  of  constructing  independently  of 
analysis  a  geometry  involving  imaginary  elements.  Thus  ac- 
cording to  Von  Staudt:  Two  conjugate  imaginary  points  may 
always  be  considered  as  the  double  points  of  an  (elliptic)  involu- 
tion on  a  real  line;  and  as  (in  analysis)  we  pass  from  an  imaginary 
number  to  its  conjugate  by  changing  i  to  —  i,  so  (in  geometry) 
we  may  distinguish  the  two  imaginary  points  by  associating 
them  respectively  with  the  two  senses  of  the  line. 

Now  the  most  essential  or  characteristic  ideal  of  geometry 
is  to  render  all  configurations  visualizable,  intuitive:  the  de- 
sideratum is,  to  use  the  German,  "Anschaulichkeit."  Whilst 
Von  Staudt's  purely  projective  methods  were  theoretically  suf- 
ficient, yet  they  were  found  to  be  in  use  cumbrous  and  compli- 
cate; accordingly  we  find  the  diverging  lines  of  analysis  and 
geometry  beginning  to  change  direction  and  to  come  together. 
"From  this  moment  a  brilliant  period  opens  for  geometrical 
research  of  every  kind.     Analysts  interpret  all  their  results  and 


INTRODUCTION.  3 

set  to  work  to  translate  them  by  constructions.  Geometers 
endeavor  to  discover  in  every  question  some  general  principle — 
in  most  cases  impossible  to  prove  without  the  aid  of  analysis."^ 
This  was  but  reflecting  the  spirit  of  the  great  Monge  who  "has 
shown  from  the  outset  .  .  .  that  the  alliance  between  analysis 
and  geometry  was  useful  and  fruitful  and  that  perhaps  their 
alliance  was  a  condition  of  the  success  for  both  of  these  branches 
of  mathematics."^  For  example,  in  the  conclusion  of  his  lecture 
"On  the  Real  Shape  of  Algebraic  Curves  and  Surfaces/'  as 
interpreted  by  geometric  models  and  Riemann  surfaces,  Klein 
says:  "These  methods  give  us  the  actual  mental  image  of  the 
configuration  under  consideration,  and  this  I  consider  the  most 
essential  in  all  true  geometry."^ 

This  ideal  of  geometry  is  entirely  consistent  with  the  ideal  of 
mathematics  as  presented  by  Von  Staudt  himself:  "Indem  die 
Mathematik  darnach  strebt,  Ausnahmen  Von  Regeln  zu  be- 
seitigen  und  verschiedene  Satze  aus  einem  Gesichtspunkte  auf- 
zufassen,  wird  sie  haufig  genothigt,  Begriffe  zu  erweitern  oder 
neue  Begriffe  aufzustellen,  was  beinahe  immer  einen  Fortschritt 
in  der  Wissenschaft  bezeichnet."^ 

In  these  ideals  we  have  the  spirit  and  aim  of  the  great  program 
proposed  by  Professor  Study  in  his  lectures  and  elsewhere,^  a 
program  in  accordance  with  which  on  the  one  hand  we  are  not 
to  be  hampered  by  assumptions  regarding  reality  or  non-reality 
and  on  the  other  hand  all  configurations  whether  real  or  imaginary 
are  to  receive  intuitive  representation. 

In  his  work  "Vorlesungen  iiber  ausgewahlte  Gegenstiinde  der 
Geometric,  erstes  Heft :  Ebene  Analytische  Kurven  und  zu  ihnen 
gehorige  Abbildungen"  Study  has  blazed  a  path  through  the 
great  domain  contemplated  by  his  program.     My  aim  in  this 

iDarboux:  "A  Study  of  the  Development  of  Geometric  Methods,"  Con- 
gress of  Arts  and  Science,  St.  Louis,  1904. 

2  Darboux,  supra. 

3  The  Evanston  Colloquium.     Lectures  on  Mathematics,  Lecture  IV. 
^  "Beitriige  zur  Geometrie  der  Lage,"  Vorwort. 

6 "  Vorlesungen  liber  ausgewahlte  Gegenstande  der  Geometrie"  (1911), 
"Zur  Differential-geometrie  der  analytischen  Curven,"  and  "Die  naturlichen 
Gleichungen  der  analytischen  Curven  im  Euclidischen  Raume,"  Trans.  Am. 
Math.  Soc,  vols.  10,  11. 


4  COMPLEX   CONICS   AXD  THEIR  REAL  REPRESENTATION. 

paper,  as  indicated  by  its  title,  is  to  apply  his  general  method 
to  an  interesting  and  important  detail. 

2.  The  Laguerre-Study  Representation  of  the  Imaginary. — There 
is  a  great  variety  of  ways  of  representing  the  imaginary  element 
by  a  real  figure.^  The  most  efficient  for  purposes  of  analysis  is 
that  due  to  Laguerre-  extended  and  developed  by  Study .^ 

The  two  families  of  minimal  lines,  right-  and  left-sided,  to 
use  Study's  term,  have  for  equations 

^  +  ^77  =  const,     and     ^  —  i-q  —  const. 

Where  ^  and  t)  are  the  rectangular  cartesian  coordinates  of  the 
00*  finite  complex  points  in  a  projective  plane.  These  lines 
through  a  point  (^',  -q')  of  the  plane  have  for  equations: 

s  +  'i'n  =  ^'  +  W     and     ^  —  i-q  =  ^'  —  ir]'. 

On  each  of  these  lines  there  is  one  and  only  one  real  point. 
Taking  {x' ,  y')  for  the  coordinates  of  the  real  point  on  the  second 
and  {u',  v')  those  for  the  real  point  on  the  first,  we  have: 

t'  +  irj'  =  m'  +  iv'     and     ^'  -  irj'  =  .r'  -  iy'. 

Considering  ?/  +  iv'  and  x'  +  iy'  as  geometric  pictures  of  two 
gaussian  numbers  w'  and  z'  we  write 

(I)  ^'  +  iv'  =  «''     and     ^'  -  iv'  =  z' , 

where  z'  =  x'  —  iy' ,  the  conjugate  of  %' .  It  is  agreed  to  take 
the  two  real  points  s'  and  ?r'  of  the  two  gaussian  planes  as  the 
Real  Representation  of  the  complex  point  (^',  77')  of  the  cartesian 
plane.  Study  symbolizes  this  representation  by  z  — >  w,  and 
calls  it  the  First  Picture  (das  erste  Bild)  of  the  imaginary  point. 
Again  we  have  for  the  conjugate  point  (|',  rj')  of  (^',  77') 

.  _  ,-,  =1'.  i-^',  I  consequently  |  ,,  _  ._,  ^  _,_ 


Thus  the  picture  of  (4',  77')  is  w' 


r.' 


1  Encyklopadie  der  Mathematischen  Wissenschaften,  III,  AD,  4o,  13-16. 

2  Oeuvrcs,  Tome  II,  pp.  89-98. 
'  "Vorlesungen,"  p.  9. 


INTRODUCTION.  5 

Hence  by  virtue  of  relation  (I)  a  perfect  correspondence  is  set 
up  between  the  totality  of  finite  complex  points  of  the  plane  and 
the  totality  of  finite  real  point-pairs:  to  any  complex  point 
corresponds  uniquely  a  real  point-pair,  and  vice  versa.^ 

The  cartesian  plane  (^,  rj)  and  the  picture  planes  (s)  and  {w) 
may  be  considered  superposed  or  not;  in  either  case  they  are  to 
be  considered  distinct.  The  oo^  real  points  of  the  plane  have 
IV  =  z;  that  is,  if  the  planes  are  considered  coincident,  these 
points  are  their  own  pictures. 

1  "Vorlesungen,"  p.  10. 


CHAPTER  I. 
REAL  CONICS. 
3.  Some  Simple  Cases. — If  we  have  given  an  equation, 

<pa,  V)  =  0, 

^,  7]    being  rectangular  cartesian    coordinates,  there  is  simul- 
taneously given,  by  virtue  of  the  relations 


^  +  iv  =  w, 

^  -  irj  =  z 

(1)                or        |="  +  ^ 

w  —  z 

^  -     2i     ' 

a  corresponding  equation 

fiio, 

z) 

=  0, 

setting  up  a  correspondence  between  the  two  picture  planes. 
This  latter  equation,  being  a  relation  between  two  complex 
variables,  expresses  a  conformal  transformation  between  the 
two  planes.  Since  one  of  the  variables  appears  in  its  conjugate 
form,  angles  under  the  transformation,  while  preserved  in 
magnitude,  are  reversed  in  sense.  Such  a  transformation  Study 
calls  an  improper  (uneigentlich)  conformal  transformation.  We 
shall  refer  to  such  a  transformation  as  reverse  conformal. 

If  the  coefficients  of  (p  are  real  the  curve  is  called  real,  although 
it  may  contain  no  real  points.  Compare  §  4  of  this  chapter, 
and  Study,  p.  46. 

Accordingly,  for  the  sake  of  completeness  and  reference,  we 
state  some  simple  cases  without  giving  the  proof,  and  develop 
others. 

(A)  The  picture  of  the  oo^  pQi^^s  on  a  real  line,  a^  +  6r;  +  c  =  0, 
a,  h,  c  being  real,^  is  the  ensemble  of  point  pairs  symmetric  (the 

1  We  shall  observe  the  following  notation  throughout  this  paper:  Complex 
quantities  shall  be  represented  by  Greek  characters,  a,  X,  |,  etc.,  and  real 
quantities  by  Latin  characters  a,  I,  x,  etc.,  with  the  single  exception  w  and  z; 
these  beiBg  complex  and  equivalent  to  u  +  iv  and  x  -{-  iy  respectively. 


REAL  CONICS.  7 

one  the  reflection  of  the  other)  with  respect  to  the  real  branch 
of  the  line.     (Study,  p.  25.) 

(B)  The  picture  of  the  oo  2  points  on  a  real  circle  is  the  ensemble 
of  point-pairs  symmetric  (geometrically  inverse)  with  respect  to 
the  circle.     (Study,  p.  36.) 

Study  gives  the  real  ellipse  as  an  example  in  his  lectures  on 
this  subject.  I  shall,  however,  consider  this  case  along  with  the 
real  hyperbola  and  real  parabola,  the  method  used  differing 
from  Study's  in  no  essential  respect. 

4.  The  Real  Ellipse. — We  have  two  cases,  (1)  where  the  ellipse 
has  a  real  branch  or  arc,  (2)  where  the  ellipse  has  no  real  branch 
or  arc.^ 

The  equations  corresponding  to  these  two  cases  are  of  course 

£2  2 

(1)  §+^=±1,         a>b. 

If  we  write  e  =  1  or  i  then  e^  =  ±  1  and  we  have 

fc2  „2 

(2)  h+l-^'- 

Form  the  pencil  of  lines 

(3)  r;=-r(^  +  ea),         t  =  s -\- it. 

This  line  has  one  fixed  intersection  with  the  ellipse,  (—  ea,  0), 
and  one  free  intersection.  The  range  of  r  through  the  domain 
of  real  numbers,  s  arbitrary,  t  =  0,  gives  the  00^  real  points  on 
the  ellipse  e  =  1.  The  same  range  of  r  gives  00^  complex  points 
of  the  ellipse  e  =  i.  The  complete  variation  of  r,  s  and  t  assum- 
ing all  possible  real  values,  gives  the  <x>-  points  of  the  conic. 

Expressing  ^  and  77  in  terms  of  the  parameter  r,  using  equa- 
tions (2)  and  (3), 

1  -  T-  2ea  2ebT 

(4)  ^  =  €a^-^py.,         ^  +  ^a  =  Y-^2-^         ^=r+72- 

1  In  investigations  in  this  field,  where  a  curve  is  considered  as  consisting  of 
00  2  points,  a  configuration  consisting  of  0=^  points  is  called  by  Segre  filo,  which 
Study  translates,  Faden,  that  is  thread.  The  difference,  then,  in  these  two 
ellipses  is  that  one  has  a  real  thread  and  the  other  has  not. 


8  COMPLEX  CONICS  AND  THEIR  REAL  REPRESENTATION. 

Then  for  2c  and  z  in  terms  of  the  parameter  we  have: 

.        .  ^    o  +  ibr 

IV  =  ^  +  17]  =  -  ea  +  2e  -^r^— .,  , 

(4')  _^  .,_ 

To  simplify  the  expressions  on  the  right  hand  of  these  equations 
and  at  the  same  time  to  bring  them  to  well-known  forms  we 
change  our  parameter  bj'  means  of  a  linear  transformation, 

.t'  -  1 
Effecting  this  change  in  (4')  we  have 


€ 

w  =  2 
(5) 


if,         ,,_,       a  —  61 


The  function  |[f  +  (l/i")]  is  one  of  the  most  common  in  func- 
tion theory.  The  Riemann  surface  belonging  to  it  is  two- 
sheeted,  the  two  sheets  being  connected  along  the  branch-cut 
running  from  —  1  to  +  1.  To  concentric  circles  about  the 
origin  in  the  ^-plane  correspond  confocal  ellipses  on  the  surface. 
To  the  pencil  of  straight  lines  through  the  origin  correspond 
hyperbolas  confocal  with  the  above-mentioned  ellipses.  A  single 
ellipse  lies  entirely  in  one  sheet,  but  the  branches  of  a  single 
hyperbola  lie  half  in  one  sheet  and  half  in  the  other.  The 
Riemann  surfaces  of  ?r  and  z  defined  by  (5)  differ  from  this 
surface  in  no  essential  respect.^ 

However,  for  some  further  simplification,  and  for  uniformity 
with  cases  to  be  discussed  later,  we  make  a  second  change  of 
parameter.     We  may  write 

, ,   ,    ,   a  +  6       ,         ,  ^   ,         or  —  IP'  c^ 

(a  -  by  +  — r-  =  (a  -  by  +  , ^^  =  Tx  +  -, 

Ti  =  (a  —  b)T'. 
I  See  Lewent,  Konforme  Abbildung,  p.  63. 


REAL  CONICS. 

Where  ^  =  c?  —  Ir,  c  being  the  focal  distance.     Thus  we  have 

(0)  ,„  =  l[,,  +  f], 

where 

Ti  =  (a  -  h)T'; 
and 

z  = 
where 


5[^'+S]' 


T2  =  (a  +  6)r'. 

Certainly  the  simplest  complex  configuration  is  the  real  straight 
line,  and  the  simplest  position  of  the  real  straight  line  is  in 
coincidence  wdth  the  ^-axis,  77  =  0.  The  picture  of  the  line  in 
this  position  is  given  by  the  ensemble  of  point-pairs  connected 
by  the  relation  w  =  z  by  virtue  of  77  =  (w  —  z)/2i.  It  is 
known  in  Conformal  Geometry^  that  any  real  analytic  curve  is 
conformally  transformable  into  a  straight  line,  in  particular, 
into  the  ^-axis  (small  regions  about  a  point  of  each  curve  being 
taken).  Conversely  the  ^-axis  may  be  transformed  conformally 
to  any  analytic  curve.  The  transformation  effecting  this, 
operating  on  iv  —  z  gives  w  =  f(z)  or  more  generally 

F(z,  iv)  =  0. 

For  a  small  region  about  a  point  ^'  on  the  ^-axis  the  points 
symmetric  (in  the  Schwartzian  sense)  to  each  other  with  respect 
to  the  ^-axis  go  over  into  points  symmetric  with  respect  to  the 
arc  of  the  analytic  curve  through  the  transform  of  ^'.  We  shall 
give  this  principle  prominent  place  in  this  paper.  And,  since  our 
transformations  are  of  the  simplest  sort,  consisting  only  of  trans- 
lations, rotations  and  inversions,  the  principle  just  stated  holds 
in  the  large,  that  is,  throughout  the  finite  portion  of  the  plane. 

The  parameter  r  is  complex  and  we  ascertain  first  the  corre- 
spondence between  the  r-plane  and  the  r'-plane,  the  two  vari- 
ables being  connected  by  the  relation: 

.r'-  1 

^Kasner,  "Conformal  Geometry,"  International  Congress  of  Mathe- 
maticians, Cambridge,  1912. 


10 


COMPLEX   CONICS  AND  THEIR  REAL  REPRESEXTATIOX. 


Solving  for  t'  and  calling  for  the  moment  t  =  x  -^  iy  and 
t'  =  re'  +  iy'  we  have,  following  the  usual  method  of  Holz- 
miiller/ 

.  ,  X  -\-i(y  -\-  1) 

x'  -h  ly'  =  - 


and 


X   —  ly   =  — 


+  /  = 


X  +  i{y  -  1) ' 

a-  -  i{y  +  1) 
X  -  i{y  -  1) ' 

x''+(y  +  ly 
x'  +{y-  ly 


Where  jJ  and  q  are  radii  vectores  from  the  points  —  i  and  +  i. 
The  variation  of  j)  and  7  gives  the  pencil  of  circles  with  ±  i  as 
limiting  points.  The  real  axis  of  r  is  given  hy  p  =  q  and  since 
r  =  piq  =  ij  we  see  that  to  the  real  axis  of  r  corresponds  the 
unit  circle  of  r'.  Further,  to  circles  of  the  pencil  in  the  upper 
half  of  the  r-plane,  p  >  q  and  hence  r  =  p/q  >  1  correspond 
concentric  circles  about  tlie  origin  of  the  r'-plane  outside  the 
unit  circle,  that  is,  the  upper  half  of  the  r-plane  maps  into  the 
outside  of  the  imit  circle,  and  consequently  the  lower  half  of  the 
r-plane  maps  into  the  region  inside  the  unit  circle  of  the  r'-plane. 
It  is  eas}^  also  to  show  that  the  imaginary  axis  of  the  r-plane  goes 
over  into  the  real  axis  of  the  r'-plane. 

Returning  now  to  equations  (6)  let  us  write 


(7) 


w  =  ecTi 
and    s  =  6(72 


where 


1  /  c"  \ 

0-1  =  -  I  ri  +  —  j  ,         Ti  =  (a  -  6)r'; 

0-2  =  :^  I  To  +  —  1  ,          r2  =  (a  +  o)T  . 


Let  (Tk  =  Sk'  +  isf:"  and  n-  =  tk  +  itk";  U>^  =  1,  2).  Separating 
the  last  equations  of  (7)  into  their  real  and  pure  imaginary  parts 
we  have 

(8)  ..'      "'        "' 


''=l[r;5TI^^+i] 


and 


*  "Theorie  der  Isogonalen  Verwandschaften." 


REAL  CONICS.  H 


Sk'    = 


To   circles   about  the   origin   in   tlie   Xft-plane,   tk'^  +  th""^  =  r*, 
correspond  ellipses  whose  equations  are 


(7-)  (7-'J 


To  lines  through  the  origin  of  the  r^t-plane,  tk"  =  mtk,  correspond 
hyperbolas  whose  equations  are 

(10)  -4^  -  -4^  =  1. 

1  +  m^      1  +  m2 
These  ellipses  and  hyperbolas  are  confocal  for 


K7-T-i(f-0^= 


and 


1  +  m2  ^  1  +  w2 

c  being  the  focal  distance. 

If  equations  (9)  and  (10)  be  expressed  in  terms  of  a  —  6  and 
a  +  6,  that  is,  if  cri  and  at  be  expressed  as  functions  of  r'  the  re- 
sults are,  writing  a'  for  a  +  6  and  h'  for  a  —  b, 


4*1''  4si' 


(90 


(^+-)  (^--y 


=1, 


(7  +  »'rJ      (7 -'*''•)' 

/2  //2 

1  +  m^       I  -{-  VI- 

These  equations  show  that  to  any  circle  of  radius  r  in  the  t'- 
plane  in  general  correspond  different  ellipses  in  the  (Ty  and 


12 


COMPLEX   COXICS  AND  THEIR  REAL  REPRESENTATION. 


(r2-surfaces.     There  is  one  exception,  namely,   r  =  1.     In  this 
case  each  of  the  equations  (9')  becomes 


(11) 


.V'         s" 


To  any  line  in  the  r'-plane  through  the  origin  correspond  the 
same  hyperbolas,  but  we  arc  not  to  understand  that  as  r'  de- 


scribes any  given  line  in  its  plane  <ti  and  o-o  describe  the  branches 
of  the  corresponding  hyperbola  together.  A  little  examination 
shows  that  they  never  move  in  the  same  direction  on  the  hyper- 
bola but  always  in  opposite  directions,  passing  each  other  on 
the  common  ellipse  (11)  of  the  two  planes.  We  do  not  carry 
this  investigation  beyond  this  point.  Further  details  are 
mentioned  by  Study,  pp.  98-103. 


REAL  CONICS.  13 

If  our  ellipse  has  a  real  branch,  that  is,  if  e  =  1  the  ic-  and 
2-surfaces  are  identical  with  the  cti-  and  o'2-surfaces.  If 
€  =  i,  the  iv-  and  s-surfaces  are  got  by  rotating  the  ai-  and 
or2-surfaces  through  90°  and  —  90°  respectively. 

We  have  shown  in  this  section  that  the  real  axis  of  the  para- 
meter plane  (r)  maps  into  the  real  branch  of  the  ellipse  €  =  1, 
while  for  the  ellipse  e  =  i  the  real  axis  of  r  maps  into  the  real 
branch  of  the  ellipse  e  =  1  turned  through  90°.  The  picture- 
points  z  and  w  are  united  in  position  for  the  ellipse  e  =  1,  and 
are  diametrically  opposite  for  the  ellipse  e  =  i.  The  elliptic 
pencil  of  circles  with  limiting  points  ±  i  have  been  shown  to 
map  into  confocal  ellipses,  circles  symmetric  with  respect  to 
the  real  axis  in  the  r-plane  go  into  ellipses  symmetric  with 
respect  to  the  real  branch  of  the  ellipse  e  =  1,  and  for  the  case 
€  =  i  symmetric  with  respect  to  the  ellipse  e  =  I  turned  through 
90°.  It  may  be  shown  that  the  hyperbolic  pencil  orthogonal  to 
the  elliptic  pencil  maps  into  the  confocal  hyperbolas.^  Corre- 
sponding curves  and  points  are  shown  in  the  figures. 

5.   The  Real  Hyperbola. — 


2      1.2  —  ^>        a  >  b. 


Introducing  the  parameter  in  the  same  way  as  in  the  case  of 
the  ellipse  we  have: 

1  +  T^  ,       2« 


26r 


V  = 


Hence 


1  -  T^' 

w  =  —  a  -\-  2 


a  -f-  ibr 


a  +  ibr 
z  =  -  a  +  2- =;r. 

1    —   T 

Changing  the  parameter  by  means  of  the  linear  transformation: 

r'  -  1 
"■  ~  r'  +  1 ' 
*  Holzmiiller,  p.  63. 


14         COMPLEX  COXICS  AXD  THEIR  REAL  REPRESENTATION. 

we  have, 


w 


(1) 


Or,  as  in  the  case  of  the  ellipse,  if  we  put 

(2)     T^  =  {a  +  ih)T'     and     to  =  (a  -  ih)r',     and     ar-\-h'^  =  &, 

c  being  the  focal  distance,  we  have. 


(3) 


W^+t]' 


^t'  =  2  I  'Ti  +  T 


\b4^- 


When  the  equations  of  (3)  are  resolved  into  their  real  and  pure 
imaginary  parts  we  have,  as  in  the  preceding  case: 


(4) 


Since  ti  =  (a  +  ih)r'  and  r-i  =  (a  —  f6)r',  the  configuration  of 
the  r'-plane  consisting  of  concentric  circles  and  radial  lines,  is 
not  altered,  lines  and  circles  merely  going  over  into  lines  and 
circles.  In  particular  the  real  axis  of  r'  goes  into  the  line  with 
slope  hja  in  the  ri-plane,  and  the  line  with  slope  —  {bja)  in  the 
To-plane.  Since  the  expansion  |  a  +  i6 1  is  the  same  for  both 
Ti  and  T2,  a  circle  of  r'  goes  into  equal  circles  in  the  ti-  and  T2- 
planes.  To  concentric  circles  and  radial  lines  of  the  ti-  and 
To-planes  there  correspond    ellipses  and  hyperbolas  in  the  W' 


REAL  CONICS.  15 

and  s-surfaces.     Their  equations  are: 


(7-)      (7-0^ 


and 


■""    /^2  \2  ^»  ^2  „2^  2       ~    Aj 


(7-)      (7-^J 


W2 

^2 

C2           ~ 

^m\- 

a;2 

1  +mi2 
2/^ 

c2 

(?m<^ 

1   +  W2^  1   +  W2^ 


where 

tan~^  m\  =  tan~^  m  +  tan~^  -  :  tan~^  iiio  =  tan~^  m  —  tan~^  -  , 

a  a 

m  being  the  slope  of  a  line  through  the  origin  in  the  r'-plane. 
The  ellipses  correspond,  but  the  hyperbolas  are,  in  general, 
different,  the  exception  being  where  m\  =  b/a  and  m2  =  —  b/a. 
For  this  case  the  above  equations  reduce  to: 

a^      0-  a^      b^ 

Now  the  transformation  t  =  (t'  —  1)/(t'  +  1)  converts  the  real 
axis  of  r  into  the  real  axis  of  r',  and  as  we  noted  the  lines  of 
slope  b/a  and  —  b/a  in  the  ri-  and  T2-planes  correspond  to  the 
real  axis  of  t'  and  hence  to  the  real  axis  of  r.  So  the  real  axis 
of  r  maps  into  the  real  branch  of  the  hyperbola. 

It  can  be  shown^  that  to  concentric  circles  about  the  origin 
in  (r')  correspond  the  pencil  of  circles  having  ±  1  for  their 
limiting  points,  and  to  the  radial  lines  in  the  r'-plane  correspond 
the  pencil  of  circles  with  vertices  ±1.  So  we  have  the  case 
corresponding  to  that  of  the  ellipse:  the  hyperbolic  pencil  of 
circles  in  the  r-plane  with  the  common  points  ±  1  maps  into 
confocal  hyperbolas  in  the  w-  and  s-surfaces;  in  particular,  the 
radical  axis  of  the  pencil  going  into  the  real  branch  of  the  hyper- 
bola. The  orthogonal  pencil  with  ±  1  as  limiting  points  goes 
into  the  confocal  ellipses. 

6.   The  Real  Parabola. — We  take  for  the  equation  of  the  para- 
bola 
(1)  V'  =  4p(^  +  P), 

^  See  Holzmiiller,  p.  63. 


IG         COMPLEX    CONICS   AND  THEIR  REAL  REPRESENTATION. 

with  the  finite  focus  at  the  origin.     Form  the  pencil, 

77  =  2Ta  +  p). 


Then, 
and, 


1  —  T-  p  2p 


,       1  +  2ir  ,       1  +  2ir 


As  in  the  previous  cases  we  simplify  these  expressions  by  a  linear 
transformation  of  the  parameter 

(2)  r  = .  for  w,  and  t  = , — ;  for  s. 

Ti  —  ^  T2  -{-  I 

Thus  we  get 

(3)  W   =   pTi^,  S   =  2^7^2". 

The  Riemann  surface  belonging  to  pr-  consists  of  two  sheets 
joined  along  the  positive  real  axis.  To  lines  parallel  to  the  axis 
of  reals  h"  =  const,  correspond  confocal  parabolas, 

To  lines  parallel  to  the  imaginary  axis,  ti   =  const.,  correspond 
the  orthogonal  trajectories  of  the  preceding  set,  namely, 

1)2  =  -  4pt,'^(u  -  ph'-). 

These  two  families   of  parabolas  are   confocal,   having   the 
origin  for  their  finite  focus. 
We  get  similar  equations  for  s, 

if  =  ^ph"\x  +  ph"^), 
if  =  -  4pt/\x  -  ptof). 

Solving  (2)  for  ti  and  t-z  we  may  write: 

n  =  ^-^,         a- =  1,2) 
the  upper  sign  going  with  ri  and  the  lower  with  T2.     Separating 


REAL   CONICS.  17 

this  equation  into  its  real  and  pure  imaginary  parts  we  have 


To  Hnes  parallel  to  the  real  axis  of  Xk,  U"  =  b,  correspond  a 
parabolic  pencil  of  circles  through  the  origin  with  centers  on  the 
imaginary  axis,  (6  T  l){t'"  +  t"")  ±  t"  =  0.  In  particular  to 
the  lines  4"  =  ±  1  corresponds  t"  =  0,  the  real  axis  of  r.  Again, 
to  lines  parallel  to  the  imaginary  axis,  4'  =  a,  corresponds  a 
second  parabolic  pencil  of  circles  through  the  origin  with  centers 
on  the  real  axis,  the  orthogonal  trajectories  of  the  preceding 
pencil.     The  equation  of  this  last  pencil  is  a{t'^  +  t"^)  —  i'  =  0. 

To  the  real  axis  of  r  then  corresponds  in  the  w-  and  s-surfaces 
the  curves  t'^  =  Ayi^u  +  p)  and  y-  =  ^y{x  +  p),  that  is,  the  real 
branch  of  our  parabola.  As  r  describes  a  circle  of  radius  r,  in 
its  upper  half-plane,  say,  r  describes  a  congruent  circle  in  the 
lower  half-plane.  Correspondingly  n  describes  a  straight  line 
t\"  =  (l/2r)  +  1  in  its  plane  and  r-i  describes  a  straight  line 
h"  =  iXl^r)  —  1,  and  in  turn  w  and  z  describe  the  confocal 
parabolas : 

»— 4p(i+iy[«+p(^-+iy], 

As  T  describes  a  circle  of  radius  r'  of  the  orthogonal  pencil,  t 
describes  the  same  circle  and  w  and  2;  describe  the  same  parabola 


^2  ^  _  P 
r 


''V      4r'V' 


Both  families  of  parabolas  are  double  decked,  that  is,  both 
sheets  of  the  Riemann  surface  are  filled  with  parabolas.  Those 
extending  infinitely  to  the  right  have  both  branches  in  the  same 
sheet.  Those  extending  infinitely  to  the  left  change  sheets  over 
the  branch-cut  running  from  the  origin  to  infinity  positively. 


CHAPTER  11. 

THE  COMPLEX  CONIC,  PRELIMINARY  CONSIDERATIONS. 

While  the  purpose  of  this  paper  is  the  reduction  of  the  equation 
and  the  real  representation  of  the  imaginary  conic,  yet  we  shall 
find  it  advantageous  to  spend  some  time  with  the  more  elementary 
configurations.  As  a  result  should  come  a  proper  orientation 
in  the  field  of  "geometry  in  the  domain  of  the  complex";  also 
suggestions  as  to  methods  of  procedure  in  the  question  proper. 
Accordingly  we  give  our  attention  first  to 

The  Complex  Line. 

7.  Relation  of  the  Pictures  of  a  Complex  Line  and  Its  Conjugate, 
or  the  Group  Property  of  the  Transformations  Belonging  to  the 
Complex  Line  and  Its  Conjugate. — The  general  equation  of  the 
complex  line  is 

A  :  a^  +  /377  +  7  =  0, 

where  ^,  t],  a,  13,  y  have  the  form  wi  +  ino  and  the  ratios  a  :  fi  :  y 
are  not  all  real.  This  geometric  configuration  has  oo-  complex 
points.^  There  are  no  conjugate  pairs  of  points  on  the  line,  with 
one  exception,  namely,  the  real  point  of  the  line  which  is  its  own 
conjugate. 

This  fact  leads  us  to  consider  in  connection  with  A  its  conju- 
gate, 

A  :  a^  +  /377  +  7  =  0. 

A  is  the  "locus"  of  the  conjugate  of  the  points  of  A,  and  con- 
versely. The  two  loci  intersect  in  their  common,  self-conjugate 
real  point. 

The  corresponding  reverse  conformal  transformations  picturing 
the  imaginary  points  of  A  and  A  are,  by  virtue  of  the  relations 
§2,(1) 

A  :  a^  +  /3r;  +  7  =  0 ;  T  :  niv  -}-  vz  -\-  2y  =  0, 

(!)-_-_  _         _  _ 

A  :  o;^  +  iSt?  +  7  =  0;         T'  :  vw  +  fjiz  +  2y  =  0, 

where  jx  =  a  —  ijS  and  p  =  a  -{-  zjS. 

'  Hereafter  the  term  point  will  mean  complex  point. 

18 


THE   COMPLEX   CONIC.  19 

Let  us  now  for  the  moment  regard  the  planes  (w)  and  (z) 
as  coincident.     Applying  T  to  a  point  z'  we  have 

(2)  fjiiv'  +  vz'  +  27  =  0, 
then  applying  T'  to  to' 

(3)  vw"  +  '^iw'  +  27  =  0. 

On  elimination  of  w'  between  (3)  and  the  conjugate  of  (2), 

'iiw'  +  vz'  +  27  =  0, 
there  results 

w"  =  z'. 

The  result  is  the  same  if  we  reverse  the  order  of  application. 
Hence  the  theorem: 

The  reverse  conformal  transformations  belonging  to  a  line  and 

its  conjugate  applied  consecutively  leave  the  points  of  the  plane  in 

place : 

T'  =  'r~^  TT'  =  T'  T  z=  ^ 

In  case  the  line  is  real  a  =  a,  j8  =  6,  7  =  c,  n  =  a  —  ih, 

V  =  a  +  ib  =]!  then  T  =  T'  :  fxiv  -\-  Jlz  -^  2c  =  0,  that  is,  the 

real  line  is  its  own  conjugate  and  further  TT'  =  T^  =  1,  the 
transformation  belonging  to  it  is  involutorial. 

8.  Reduction  of  the  Equatioti  of  the  Complex  Line  to  Canonical 
Form. — Putting  A  and  A  in  the  form  R  -\-  iT,  R,  T  being  linear 
functions  in  $.  77  with  real  coefficients,  we  have 

A,  A  :  ii  ±  iL2  :  ai^  +  birj  +  Ci  ±  i(a2^  +  b-zv  +  C2)  =  0. 

Thus  A  and  A  are  identified  as  members  of  a  complex  pencil 

n  :  Zi  +  KL2  =  0, 

where  k  =  k'  -\-  ik" .  Belonging  to  this  pencil  there  is  of 
course  a  single  infinity  of  real  lines,  k'  arbitrary,  k"  zero.  On 
each  value  of  k',  k/  say,  there  is  built  up  a  single  infinity  of 
imaginary  lines,  k/'  arbitrary.  We  may  thus  distribute  the 
double  infinity  of  complex  lines  into  a  single  infinity  of  sub- 
pencils  each  with  real  bases  and  each  containing  two  real  lines, 
the  bases,  k'  =  k/,  and  00,  and  a  single  infinity  of  imaginary 
lines,  k"  arbitrary.  According  to  this  classification  A  and  A 
belong  to  the  sub-pencil  k'  =  0  with  bases  Li  and  L2. 


20         COMPLEX  CONICS  AND  THEIR  REAL  REPRESENTATION. 

We  proceed  now  to  simplify  the  equations  of  A  and  A  by 
referring  them  to  other  bases,  namely  a  certain  rectangular 
pair.  Assuming  ai&2  —  azbi  4=  0  and  aia2  +  &1&2  4=  0  ^  A  and  A 
have  their  common  real  point  finitely  located.  We  translate  the 
origin  to  this  point  and  at  the  same  time  write  A  in  the  so-called 
normal  form  of  elementary  geometry, 

a  B 

^^     ^  +     ,  V  =  0 

or 

^  cos  d  -\-  7}  sin  6  =  0 
where 

cos  6  =  cos  (s  +  it)  =  cosh  t  cos  s  —  i  sinh  t  sin  s, 
and 

sin  6  =  sin  (s  +  it)  —  cosh  i  sin  s  +  i  sinh  t  cos  s. 

Thus  the  normal  forms  of  A  and  A  are 

(1)        ^  cos  5  -}-  77  sin  s  ±  i  tanh  t{^  sin  s  —  rj  cos  s)  =  0. 

Functions  of  the  angle  6,  or  of  its  component  parts  are  furnished 

by  the  relations 

bi  +  62  tan  q  Oi  tan  q  —  a^ 

a\  +  0,2  tan  9  61  tan  q  —  h^. 

and 

«i  —  ^2  cot  g'  61  +  &2  cot  q 


tanh  ^  = 


61  —  62  tan  g  ai  +  02  tan  g' ' 


where  q  =  ^  arc  (a^  +  /3^).     The  sign  of  Va^  +  |3-  is  that  required 
when  a  and  /3  are  real. 

Equation  (1)  shows  A  and  A  referred  to  new  bases,  two  per- 
pendicular lines  belonging  to  the  pencil  11.  These  new  bases 
are  connected  by  the  relation:^ 

n  :  L'  cosh  t  +  XL"  sinh  i  =  (1  +  \){Li  +  kU), 
where 

L'  =  ^  cos  5  +  77  sin  s, 

L"  =  ^  sin  s  —  T]  cos  s. 

Taking  U  and  L"  as  coordinate  axes  our  equation  of  the  com- 

1  If  0102  +  ^162  =  0,  Li  and  L^  are  perpendicular  to  each  other  and  the  second 
transformation  worked  out  above  is  not  necessary. 

-  Newenglowski,  "Cours  do  Geometric  Analytique,"  sec.  446,  old  ed. 


THE  COMPLEX   CONIC,  21 

plex  line  reduces  to  the  form  •} 

A,  A  :  77  =  zhi  tanh  t  ^. 

9.  Real  Represcntatioji  of  the  Complex  Line. — We  have  thus 
reduced  the  equation  of  the  complex  line  containing  two  effective 
complex  coefficients  to  one  in  which  there  is  but  a  single  coef- 
ficient and  it  is  pure  imaginary  in  form.  Writing  A  and  A 
separately  with  their  accompanying  transformations  we  have 

A  :  7]  =  i  tanh  t^;         T  :  iv  =  e~-'2, 
A  :  77  =  -  i  tanht^;  T'  :  lo  =  e^^z. 

The  limits  of  t  and  tanh  ^are  —  co<^<  +  oo,—  1<  tanh  t 
<  +  1.  If  we  allow  t  to  become  infinite  and  assume  its  upper 
and  lower  bounds  then  we  have  the  minimal  pair  of  the  pencil, 

^  +  iv  =  0, 
^  -  *r7  =  0. 

Since  e-'  and  e~-'  are  real  the  picture-pairs  z  and  w  are  observed 
to  lie  on  rays  symmetric  with  respect  to  the  first  bisector  of 
A  and  A. 

If  we  introduce  with  Study,  page  52,  a  parameter  p  measuring 
the  distance  from  the  vertex  of  the  pencil  to  the  points  on  A 
we  have 

^  =  p  cos  it  =  p  cosh  t,         w  =  —,  p, 
(2)  H  ,  ^ty> 

77  =  p  sin  it  =  ip  sinh  t,         z  =  e'p- 

Since  wz  =\p\^,  for  any  p,  to  and  z  are  seen  to  be  inverse  points 
with  respect  to  the  circle  of  radius  \p\,  tv  having  the  same 
argument  as  p,  and  z  its  negative. 

The  Complex  Circle. 

In  a  manner  similar  to  that  used  in  the  case  of  the  line  let  us 
take  up  the  case  of  the  circle. 

10.  Group  Property  of  Inversion  loith  Respect  to  a  Complex 
Circle  and  its  Conjugate. — The  equation  of  a  complex  circle  and 

1  Cf.  Study,  pp.  29,  52. 


22         COMPLEX   CONICS   AND  THEIR  REAL  REPRESENTATION. 

its  conjugate  with  their  accompanying  transformations  are 

K:e  +  r  +  2«^  +  2^77  +  7  =  0, 
T  :  wz  -\-  fxw  ■{■  vz  +  y  =  0, 

K  :  ^2  _^  7j2  +  2a^  +  2^77  +  7  =  0, 
T'  :  icz  +  ^zo  +  /xl  +  7  =  0, 

where   ^,   t],  a,    •  •  •    are  complex  quantities  and  n  =  a  —  i^, 

V  =  a  -\-  i0.  Neither  T  nor  T'  is  involutoric,  but,  as  before, 
applying  T  to  a  point  z'  of  (z) 

(1)  w'z'  +  ixw'  +  ^r  +  7  =  0; 
and  applying  T'  to  iv'  of  (iv) 

(2)  iv"w'  +  pw"  +  Jiw'  +  7  =  0, 
then  subtracting  from  (2)  the  conjugate  of  (1)  we  get 

(f/  +  v)iiv"  -  z')  =  0. 

Obviously  iv'  cannot  equal  —  v  for  all  values  of  z',  hence  w"  =  2' 
and  r  =  T-\  TT'  =  1. 

In  this  case  and  in  the  preceding  we  might  have  expressed  w 
explicitly  in  terms  of  z  and  arrived  at  the  same  result  by  sub- 
stitution. This  relation  is  also  evident  by  a  mere  examination 
of  the  two  expressions  T  and  J",  considering  in  the  one  z  as  the 
independent  variable  and  in  the  other  iv  as  the  independent 
variable.  When  a  =  a,  ^  =  b  and  7  =  c  that  is,  when  the 
circle  is  real  (in  the  sense  of  Segre)  we  have  n  =  a  —  ib  and 

V  =  a  -\-  ib  =  jjL,  so  T  ^  T'  :  wz  -{-  fxw  -\-  fxz  -\-  c  =  0. 

11.  Reduction  of  Equations  of  K  and  K  to  Canonical  Form. — 
Breaking  up  K  and  K  into  their  real  and  pure  imaginary  parts 
we  have 

K,  K  :  ^2  +  77-  +  2a,^  +  26177  +  Ci  ±  i{2a2^  +  26077  +  Co)  =  0. 

They  are  thus  seen  to  be  members  of  a  complex  pencil 

n  :  Ci  +  kR  =  0, 

where  Ci  =  ^^  +  r?-  +  2ai^  +  26177  +  Ci  =  0  is  a  real  circle  of 
the  pencil,  R  =  2a2^  +  26377  +  Co  =  0  is  the  radical  axis  of  the 
pencil  (also  real),  and  k  =  /c'  +  ik". 


THE   COMPLEX   CONIC.  23 

Clearly  we  may  simplify  our  configuration  by  a  change  of 
axes.  This  we  do  making  the  axis  of  centers  the  ^-axis  and  the 
radical  axis  the  77-axis.     As  a  result  we  have 


where 


and 


Il'.e  +  v'-h  2a'^  +  c'  +  2X^  =  0, 
,       2(aia2  +  6162)  -  ea  ,       ^  ,^     " 


X  =  Vas^  +  h.^K;         (to,  770), 


coordinates  of  the  new  origin. 

Among  the  real  circles  of  11  there  is  one  with  its  center  at  the 
(new)  origin  and  is  given  by  X  =  —  2a'.  Taking  this  circle 
with  the  radical  axis  as  bases  we  have 

n  :  ^2  _^  772  +  2ix^  +  c'  =  0, 

where  ^t  =  X  +  2a'.  The  character  of  the  pencil — whether 
hyperbolic,  elliptic  or  parabolic — depends  on  whether  c'  is 
greater  than,  less  than,  or  equal  zero. 

Our  original  circles,  K  and  K  referred  to  the  new  bases  are 
seen  to  be  given  by  the  values  ju  =  a'  ±  i  Vao^  +  h<f,  that  is^ 

/   ,    •    //  /      2(aia2  +  &1&2)  -  g2  „      J    ..JO 

2Va22  +  fe2' 

We  have  thus  succeeded  in  reducing  the  equation  of  the  complex 
circle 

K:e-i-T-\-2a^  +  2^rj  +  y  =  0, 

containing  three  complex  coefficients  to  a  canonical  form 

K  :  ^^  +  7,2  +  2m^  +  c  =  0, 

containing  but  a  single  complex  coefficient. 

12.  The  transformations  belonging  to  K  and  K  expressed  in  the 
canonical  form  are 

K  =  0;  T  :wz  +  ix(iv  +  2)  +  c  =  0, 

(3) 

K  =  0;         r  :  icz  +  )u(w  +  2)  +  c  =  0. 

These  transformations  of  course  are  not  involutoric  any  more 


24         COMPLEX  CONICS  AXD  THEIR  REAL  REPRESENTATION. 

than  the  original  ones  were.  So  we  cannot  consider  (2)  and 
(u')  coincident.  To  study  the  corresponding  movements  in 
(z)  and  (w)  we  introduce  a  parameter  plane  defined  in  the 
following  manner:  Setting 

77  =  0,     •  •  v,     w  =  z  in  (3), 
then 

and 

r,  r'  =  -  M  ±  Vm-  -c  =  w',  ic"  or  z',  z". 

Form  the  pencil  of  lines  7?  =  t(^  —  ^"),  t  =  s  -\-  it.     Hence 
t       t"         2p  „         2p 


and 

=    ^^^  _    //  _     ^P 

where  p  =  V/x-  —  c  =  ri  +  ii'2,  the  radius  of  the  complex  circle. 
Setting  w'  =  IV  —  iv",  z'  =  z  —  z"  and  dropping  the  primes, 

(A\                               2p                                     2p 
(4)  w  =  z r-         and         z  =  :, — . 

1   —  IT  1    —  IT 

If  r  describe  its  axis  of  reals,  t  =  0,  we  have,  on  elimination  of  s, 

(u  —  TiY  -{-  (v  —  r^y  =  r^ 
and 

{x  -  rif  +  {y  -  TiY  =  ;•-, 

where  r-  =\p\"  =  rr  +  r^^.  w  and  z  thus  are  seen  to  describe 
congruent  circles  in  their  respective  planes. 

If  T  describe  its  axis  of  pure  imaginaries,  s  =  0,  there  result 
on  the  elimination  of  t 

V  =  —II,        and        V  =  —  —x. 

Our  picture  planes  are  thus  seen  to  be  divided  into  four  regions 
corresponding  to  the  four  quadrants  of  the  parameter  plane. 
The  paired  points  of  the  two  congruent  circles  in  (s)  and  {w) 
are  the  points  which  in  the  real  case  are  united  in  position  pictur- 
ing the  00 1  real  points  of  the  circle.     We  superpose  the  planes 


THE   COMPLEX  CONIC. 


25 


now  and  notice  a  sort  of  unfolding  process  of  the  picture  planes 
due  to  the  entrance  of  the  complex  quantity.  Let  us  examine 
the  parabolic  case.     Here  we  have: 


^'   =0,  t"  =    _  9 


e-\-T  +  2m?  =  0, 

Zfj.,         p  =  n  =  n  -{-  ir2. 


Considering  the  planes  superposed,  the  origins  coincident  and 
the  axes  of  reals  of  (to)  and  (z)  coinciding  with  the  ?-axis  of  the 
cartesian  plane  we  have: 

2ju  2ifXT 


IV  =  —  2^  + 


1  —  IT      I  —  ir' 


z  =  -  2fx-\- 


2tx 


If 


If 


2^/^r 
ir      I  —  vr' 


T  =   -   00,      -  1,  0,  +1,  +  <x>, 

W  =  10^,  W-i,  0,  w+i,  w«, 

3  =  Soo,  Z—i,  0,  Z-^-i,  Zof. 

T  =  -  CO,     -  ^,  -  i^,  0,  +  ^i,      +i,      +  co, 

00 ,  Wi,  0,  -  f /z,      -  n,     w^, 


w  =  w^, 

Z    =    Zgoy 


21  on. 


Thus  having  in  mind  that  t  ->  w  is  a  direct  transformation  and 
T  — >  2  is  a  reverse  transformation,  the  angles  being  preserved 
in  sense  in  the  one  and  reversed  in  the  other,  we  shall  see  just 
what  regions  of  (z)  and  (w)  correspond.  They  are  so  indi- 
cated in  the  figure.  The  situation  is  apparent  if  we  consider 
//  as  a  sort  of  parameter  varying  in  its  pure  imaginary  part 


26         COMPLEX  CONICS  AND  THEIR  REAL  REPRESENTATION. 

only.  Let  /x  =  Wi  +  ^^2.  ??Z2  =  0  gives  a  real  circle  of  the 
pencil  with  its  center  (—  mi,  0).  The  variation  of  irii  from  0  is 
seen  to  cause  an  iinfolding  of  the  pseudo-real  pair  from  the  real 
circle,  the  centers  of  these  circles  being  {—mi,  ±  W2).  For  m^ 
negative  merely  interchanges  the  figures  z  and  w.  For  mi 
negative,  the  figures  are  reflected  over  the  77-axis.  It  is  inter- 
esting to  trace  the  path-curves  of  z  and  w  due  to  the  variation  of 
W2,  holding  T  fixed.  They  are  straight  lines  as  is  evident  when 
we  write 

2ir                                                      2iT 
10  = r-(??2i  +  ^2)        and        z  =  , ^(mi  —  imo). 

I    —    IT  1    —    IT 

In  particular  the  points  z,  w,  whose  united  positions  represent 
the  00 1  real  points  in  the  real  case  trace  the  lines 

y  =  s{x  +  2wi)         and        v  =  s{u  —  2mi). 

The  hyperbolic  and  elliptic  cases  do  not  introduce  any  essential 
differences. 

Study  has  given  the  elementary  transformations  making  up 
reflection  with  respect  to  the  complex  circle.^  In  the  light  of 
the  foregoing  considerations  we  may  state  the  following  set, 
which  is  equivalent  to  that  given  by  Study.  Calling  the  con- 
gruent circles  in  (2)  and  (w)  picturing  the  real  axis  of  the  para- 
meter, the  Congruent-Pair,  and  the  lines  picturing  the  axis  of 
pure  imaginaries,  the  Basic-Lines,  the  transformations  in  question 
are  as  follows: 

The  transformation  picturing  the  complex  points  on  an  imaginary 
circle  consists  of  three  elementary  reflections,  namely,  a  reflection 
over  the  center  axis  determined  by  the  circle  and  its  conjugate,  a 
reflection  over  one  of  the  Basic-Lines,  and  a  reflection  over  one  of 
circles  of  the  Congruent-Pair .  These  may  he  taken  in  any  order 
hut  the  Basic-Line  and  the  circle  of  the  Congruent-Pair  must  not 
helo7ig  to  the  same  planes. 

1  Page  32. 


CHAPTER  III. 

THE  COMPLEX  CONIC,   REDUCTION  OF  EQUATION. 

With  the  processes  and  results  of  the  preliminary  considerations 
in  mind  we  give  our  attention  now  to  the  subject  proper. 

13.  The  General  Equation  and  Its  Corresponding  Transforma- 
tion.— The  general  equation  of  a  complex  conic  and  its  conjugate 
are, 

T  :ae  +  2I3^V  +  7v'  +  25^  +  2er7  +  r  =  0, 

(I)  -     _  -  _  - 

T  :ae  +  2l3^ri  +  yr  +  25$  +  2ir;  +  f  =  0, 

where  all  the  quantities  entering  are  complex  and  the  equations 
are  irreducible. 

Each  configuration  contains  a  double  infinity  of  points  and 
the  conjugates  of  the  points  of  the  one  lie  on  the  other.  They 
have  four  points  in  common.  These  may  all  be  real  or  all 
imaginary,  or  any  of  the  intermediate  cases.  It  is  not  necessary 
to  make  a  separate  discussion  for  any  particular  case. 

The  transformations  picturing  the  complex  points  on  these 
"curves"  are  given  through  the  relation  §  2,  (1). 

Thus  we  have 

r  =  0; 

T  :  axw"  +  2/3iZf2  +  712=  +  25iz^  +  2€i2  +  4^  =  0, 
(II) 

r  =  0; 

T'  :  yiw""  +  2^iivz  +  ai2-  +  2iiw  +  25i2  +  4^  =  0, 
where 

ai  =  a  -  7  -  2il3,  5i  =  2(5  -  it), 

y^  =  a-y  +  2i^,         ^^  "  "  +  ^'         ei  =  2(5  +  ie). 

Each  transformation  is  (2,  2)-determined  and  hence  requires 
a  two-sheeted  Riemann  surface  for  unique  determination. 

We  note  that  when  /3i  =  0  the  variables  are  rationally  separable, 

(III)  aiio"  +  2hw  =  -  (712'  +  2eiz  +  4^). 

27 


28         COMPLEX   CONICS  AND  THEIR  REAL  REPRESENTATION. 

This  requires  that  a  -\-  y  =  0  which  corresponds  to  the  real 
case  for  an  equilateral  hyperbola.  We  give  this  case  special 
consideration,  §  18. 

If  we  take  the  conjugate  of  T  and  consider  z  the  dependent 
variable  we  observe  that  it  is  identical  with  T';  hence,  as  in  the 
previous  cases,  the  transformations  belonging  to  a  complex  conic 
and  its  conjugate  are  connected  by  the  relation  that  the  one  is  the 
inverse  of  the  other:  T'  =  T~^. 

If  the  coefficients  of  V  are  real  T  and  T'  become  identical  and 
reduce  to 

C  =  0; 

(IV) 

T  :  aiw"-  +  2biicz  +  a^z^  +  25iW  +  2hz  +  ^h  =  0. 

The  transformation  z  -^  w  picturing  the  complex  points  on  a  real 
conic  is  an  equation  of  the  second  degree  in  z  and  w\  the  coefficients 
of  the  square  terms  and  the  terms  of  first  degree  being  conjugate 
complex  in  pairs;  the  coefficients  of  the  product  term  and  the  constant 
term  being  real} 

If  further  we  consider  only  those  curves  admitting  oo^  real 
points,  we  may  put 

^  +  ir}  =  z         and         ^  —  iri  =  z. 

Our  equation  then  becomes 

(V)        T  :  axz^  +  2byzz  +  a^z"  +  2hz  +  25i2  +  4/i  =  0.^ 

Let  us  compare  the  expressions  for  the  invariants  and  the 
conditions  for  the  different  species  of  conies  in  terms  of  the 
coefficients  of  (I)  considered  as  real,  a  =  a,  ^  =  x,  •  •  •  and 
those  of  (V).     We  have 

r  =  0, 

(D 


C  =  0  •  • 

'  (V), 

) 

a  -\-  c  ■  • 

•bu 

¥-  ac  ' 

.•6i- 

\<Xl 

Hence  in  terms  of  the  coeflBcients  of  (V)  we  have  the  following 

^  Perna,  "Le  Equazione  delle  curve  in  coordinate  complesse  coniugate," 
Palermo  Rendiconti,  17  (1905),  p.  65. 

2  Cesaro,  "Sur  la  determination  des  foyers  des  coniques,"  Nouv.  Ann.,  60 
(1901),  p.  1. 


THE    COMPLEX   CONIC,   REDUCTION   OF   EQUATION.  29 

conditions  for  the  different  species  of  proper  conies  and  their 
special  cases: 

ai  =  0,  Circle, 

61  =  0,  Equilateral  hyperbola, 
Iai|  <  61,  Ellipse, 
|ai|  >  61,  Hyperbola, 
I  ail  =  ^ij  Parabola.^ 

14.  Reduction  of  the  General  Equation  to  Canonical  Form.—li 
we  write  (I)  in  the  form: 

r,  f  :  Ci  ±  iC2  =  ai^  +  261^77  +  c^i'  +  2di^  +  2^177 

+  /i  ±  i{a2e  +  2b2^r]  +  C2r  +  2d.^  +  2e.ri  +  /a)  =  0, 

we  identify  F  and  F  as  members  of  a  complex  pencil 

n  :  Ci  +  kCo  =  0,        K  =  k'  +  ik", 

F  and  F  being  given  by  /c  =  i  and  —  i  respectively. 

Among  the  real  conies,  k"  =  0,  of  the  pencil  there  is  one  and 
only  one  equilateral  hyperbola,^  for  we  have : 

(ai  +  k'a^)^  +  2(bi  +  k%)^rj  +  (ci  +  k'c^W  +  '  •  •  =  0, 

and  the  value  of  k'  which  renders  this  conic  an  equilateral 
hyperbola  is 

,  ,   _  «!  +  Cl 

^2  +  C2 

If  tti  +  Ci  =  0,  Ci  is  the  required  hyperbola  and  A;'  =  0  is  the 
value  of  the  parameter  giving  it.  If  a2  +  C2  =  0,  C2  is  the 
required  curve  and  k'  =  qo  .  If  both  ai  +  Ci  =  0  and  02  +  c-i  =  0 
both  Ci  and  C2  are  equilateral  hyperbolas  and  k'  is  indeterminate. 
In  this  case  all  the  conies  of  the  pencil  are  equilateral  hyperbolas. 
(See  §  18.) 

Supposing  the  pencil  to  have  only  one  equilateral  hyperbola* 
we  have 

H:C^  +  k'C2  =  0,        k'  =  -«-^\ 

a2  +  C2 

Taking  H  and  Ci  as  bases  we  have 

n  :  /^  +  XCi  =  (1  +  X)(Ci  +  kCo), 

^  Cesaro,  loc.  cit. 

2  Niewengloski,  "Cours  de  Geometric  Analytique,"  sec.  460. 


30        COMPLEX  COMCS  AND  THEIR  REAL  REPRESENTATION. 

where 


X  = 


k'  -_K  ,   _  fll  +  Cj    J 

,  K     — 

02  +  C2 


We  now  take  the  asymptotes  of  the  equilateral  hyperbola  as 
our  coordinate  axes.     We  have 


(1)  //  :  a'e  +  26'^77  -  aV  +  2(^'^  +  2e'r]  +/'  =  0, 

where 


a'  =  ai  +  k'a2,         h'  =  hx  +  !:%, 
or 


k'  =  - 


gi  +  Ci 

<l2   4"  C2  ' 


n   = 


ni 

712 

+ 

Wl 

W2 

|ai 

<Z2 

Ci 

C2 

L        n  =  a,  h,  c,  •••,/. 


The  transformations  effecting  this  change  of  coordinate  axes 
are, 

^  =  /r  -  mv'  +  U 

7]  =  m^'  +  W  +  r/o, 
where 

D',  E',  F'  being  the  algebraic  complements  of  d',  e',  f  in  the 
discriminant  of  //, 

I  =  yl ^ ,         m  =  y\ ^ — , 

^Si  —  52  ^*i  —  52 

Si  and  52  being  the  asymptotic  directions  of  H  furnished  by 

a'e  +  2h'^r)  -  a'jf  =  0, 

5l,  S2   =    -, . 

As  a  result  of  this  transformation  the  equation  of  H  becomes 

■^(sO,  ^o)  r,,  ,2     ,     ,,2 


(2) 


H'  :  2^v  + 


^IF'     ' 


F'  =  a'-  +  h'% 


and  Ci  becomes 


(3)  C  :  ae  +  26^^  +  cr?-  +  2d^  +  2er/  +  /  =  0, 

where 

Ci^i  +  26i  —  ai52 
a  = , 

51-52 

^  Niewenglowski,  loc.  cit.,  446. 


THE   COMPLEX   CONIC,   REDUCTION   OF   EQUATION.  31 


b  = 


c  = 


Sl  —  S2 

aiSi  —  26i  —  C1S2 

Si  —  So 


0^0   ^Si  —  S2       arjo  \5i  —  52 
^so  ^Si  —  So       driQ   y& 


S\   —So 

^0,  Vo  being  the  coordinates  of  the  new  origin. 

There  is  one  real  conic  C  of  the  pencil  IT  whose  axes  are  parallel 
to  the  asymptotes  of  H'.  This  conic  is  given  by  X  =  —  1/6, 
taking  H'  and  Ci  in  their  reduced  forms  (2)  and  (3).     Thus 

C  ^C  -hH'  =  0. 
Taking  C  and  H'  as  bases  we  have 

where 

-  1/6  +  X 
M  = ^^ . 

Thus  we  have  finally 

n  :  a^2  _|_  2jLi^7;  +  erf  +  2d^  +  2ey]  +  f  +  ixh  =  0, 
where 

h  =  p= — . 

The  values  of  the  various  parameters  giving  our  original 
conies  r  and  T  referred  to  the  different  bases  and  the  different 
configurations  of  reference  are  assembled  as  follows : 

(I)    Hi  :  Ci  +  kCs;         r,  f  :  Ci  +  k"C2  =  0,         /c"  =  ±  i, 
Uo.H  +  XCi;        r,  r  :  ^  +  V'Ci  =  0, 

±  I  02  +  <?2 

After  change  of  origin  T,f  :H'  -  X"  VFC  =  0 

Ha :  c  +  M^';       r,  f  :  C  +  fx"ir  =  0, 
(III)  ,,  _  -  1/6  +  V^VF  _  _  .    ,        1 


32         COMPLEX  CONICS   AND   THEIR  REAL  REPRESENTATION. 

We  have  thus  reduced  the  equation  of  our  conic 

(I)  ae  +  2/3^^  +  yr  +  25^  +  2e77  +  f  =  0 

containing  five  effective  constants  to  an  equivalent  one 

(VII)         ae  +  2/x^77  +  cr?2  +  2d^  +  2er]  +  f  +  fxh  =  0 

containing  only  two  complex  constants  which  are  linearly  and 
integrally  related. 

15.  "Localizing"  the  Conic  in  the  Complex  Pencil. — We  have 
identified  our  conic  as  a  member  of  a  complex  pencil  of  conies 
having  two  real  conies  (proper  or  degenerate)  as  bases.  We 
now  proceed  to  "localize"  it  among  the  double  infinity  of  conies 
of  the  pencil.     The  locus  of  centers  of  11  is  given  by 

^  =  M^  +  C77  +  e  =  0. 

Which  on  the  elimination  of  ix  yields 

Cz  :  a^-  —  err  +  ^^s  —  ct)  =  0, 

an  ellipse  or  an  hyperbola  according  as  C  is  an  hyperbola  or  an 
ellipse. 

The  double  infinity  of  points  on  Cz  are  the  centers  of  the 
double  infinity  of  conies  of  11.  The  single  infinity  of  real  points 
on  Cz  are  the  centers  of  the  single  infinity  of  real  conies  of  11. 
The  double  infinity  of  imaginary  conies  have  their  centers 
pictured  by  double  infinity  of  point-pairs  z-*  w  associated  by  a 
Schwarzian  reflection  over  the  real  branch  of  C3. 

Thus  at  this  stage  of  the  investigation  we  are  able  to  "  localize" 
our  conic  to  the  extent  of  determining  the  picture  of  the  center. 

In  general:  The  point  pairs  Zo  — »  Wq,  zvq  — >  Zq  picturing  the  center 
of  an  imaginary  conic  and  its  conjugate  are  symmetric  ivith  respect 
to  a  real  central  conic,  namely  the  locus  of  centers  of  the  real  conies 
of  a  pencil  determined  by  the  real  and  pure  imaginary  component 
parts  of  the  conic. 

Such  localization  of  the  foci  is  not  so  simple  since  they  are 
known  to  lie  on  a  bicircular  sextic. 


CHAPTER  IV. 

THE  REAL  REPRESENTATION  OF  THE  COMPLEX  CONIC. 

In  this  chapter  we  consider  the  reverse  conformal  trans- 
formation z-^iv  of  the  form  §13,  II  which  pictures  the  oo^ 
points  whose  coordinates  satisfy  an  equation  of  the  type  §  1'],  I 
or  §  14,  VII.  Or,  stated  more  exphcitly,  given  a  point  (^',  tj') 
satisfying  an  equation  of  the  above  mentioned  type,  there  is 
simultaneously  given,  by  virtue  of  the  relations  ^'  +  irj'  =  iv' 
and  ^'  +  it)'  =  z',  a  pair  of  associated  points  of  the  two  picture 
planes  (w)  and  (z).  This  point-pair  z' -*  iv'  we  call  the  real 
picture  of  the  point  (^',  r;')  and  the  ensemble  of  such  pairs 
picturing  the  double  infinity  of  points  on  the  conic  we  call  the 
Real  Representation  or  Real  Picture  of  the  complex  conic.  We 
shall  find  that  these  associated  point-pairs  may  be  grouped  in 
their  respective  planes  on  two  orthogonal  families  of  curves. 
The  two  orthogonal  nets  thus  formed  in  the  two  picture  planes 
are  more  or  less  similar  depending  on  the  complexity  of  the 
case,  being  in  the  case  of  real  conies  identical. 

We  might  take  for  the  equation  of  our  conic  §  13,  I  and  seek 
its  real  representation  through  the  corresponding  transformation 
§  13,  II.  The  method  here  developed  would  be  found  suf- 
ficient. But  we  shall  find  it  somewhat  simpler  and  more  inter- 
esting to  take  the  so-called  canonical  form  §  13,  VII — simpler 
because  we  have  only  one  complex  quantity  among  the  coef- 
ficients, and  with  the  vanishing  of  its  pure  imaginary  part  we 
have  at  once  the  real  case — interesting  because  we  shall  be  able 
at  various  points  of  the  development  to  observe  just  how  the 
entering  of  the  imaginary  affects  the  configuration. 

Accordingly  we  take  for  the  equation  of  our  conic: 

(1)     ae  +  2^^n  +  C7,2  +  2(1^  +  2e-n  +  ^  =  0,         T  =  /  +  ^f'^ 

writing  /3  =  6i  +  ibi  in  the  place  of  m  in  §  13,  VII  for  uniformity 
of  notation.     The  corresponding  equation  in  z  and  iv  given  by 

33 


34        COMPLEX  CONICS  AND  THEIR  REAL  REPRESENTATION. 

the  relation 


(2) 

is 

where 


w  =  ^  +  iri, 


or     ?  =  2  (2  +  w'), 


=  t 


irj; 


7;  =  2  (2  -  w), 


(3) 


aw-  +  2biuz  +  72-  +  2div  +  2€2  +  ^'  =  0, 
a  =  a  —  c  —  2zj3,         5  =  2(d  —  ie),  b  =  a  -\-  c, 


y  =  a-  c  +  2ifi,         e  =  2((f  +  ie)  -  6,         r'  =  4i'. 

We  shall  use  the  following  notation  for  the  discriminants  of 
(1)  and  (1')  and  the  complementary  minors  of  their  elements; 
the  symbols  in  the  first  column  in  each  table  being  for  the 
complex  case,  62  4=  0,  and  those  in  the  second  column  for  the 
real  case,  60  =  0: 

(1)  ar  +  2i3$77  +  cyf  +  2d^  +  2e7?  +  s"  =  0; 

(1')  aw^  +  2hwz  +  7|2  +  2bw  +  262  +  f '  =  0, 


a 

/3    d 

13 

c     e 

d 

e     f 

^e,  H 


c^  —  e^  =  A,  A 
(4)  de-^t^B,  B 

a^  -  d^=r,  C 
Pe  -  cd  =  A,  D 
^8  -  ae  ^E,  E 
ac  -  fi^  ^Z,  F 


a     b     8 

b     y     e     ^  X,   T, 
8      e     r' 
yr  -  e  -  21,  31', 
(4')        8e  -  6r  -  33,  33', 

ar  -  5^  -  e,  i', 

be  -  y8  =  ©,  !D', 
65  -  ae  =  e,  2)', 
«7  -  &^  =  3,  /^'. 


We  have  the  following  relations  between  the  quantities  of  the 
two  tables. 

X=  -  160,     3  =  -  4Z,  !D  =  -  4[A  +  z'E], 

(5)    21  =  -  4[A  -  r  +  2iB],     33  =  -  4[A  +  T], 

S  =  -  4[A  -  r  -  2iB],  C?  =  -  4[A  -  iE], 

Relation   (2)   may  be  looked   upon   as  a  transformation  of 


THE   REAL  REPRESENTATION   OF  THE   COMPLEX   CONIC.        35 


coordinates,  in  fact  the  quantities  z  and  w  are  called  Isotropic 
Coordinates. 

Considering  iv  and  z  as  rectangular  coordinates  equation  (2) 
expresses  an  imaginary  projective  transformation.  The  deter- 
minant of  this  transformation  is 


J  = 


=    —  9i 


,1      -  i 
We  rewrite  (5) 

(5')   51  =  J-[A  -  r  +  2iB],     33  =  riA  +  T], 

e  =  J2[A  -  r  -  2iB],  (g  =  J2[A  -  lE]. 

Thus  it  appears  that  0  and  Z  are  invariant  under  this  imag- 
inary projective  transformation  just  as  in  the  real  case. 

If  Z  4=  0,  we  have  for  the  coordinates  of  the  center  and  its 
corresponding  picture 


^0    = 


Vo  = 


j3e  —  cd  _  A 

ac  -  iS2  "  Z 

I3d-ae  _  E 

ac  -  iS-'  ~  Z  J 


Zo-^Wo 


Wo  =  lo  +  ivo  = 


20 


—  ^0  —  ^770  = 


z      ~  S' 

A-iE  _  Q 

z      ~  S- 


Here  again  we  observe  the  similarity  of  notation. 

We  proceed  now  to  a  detailed  examination  of  equation  (T). 
It  is  an  implicit  relation  between  two  complex  variables.  Solving 
for  w  we  have 

ty  =  1  [_  (62  +  5)  ±  V(62  -  ay)z'  +  2(65  -  ae)z  -^  8''  -  a'H, 
a 

or  using  the  notation  of  (4)  and  (4') 

w  =  -[-  (bz  +  5)  ±  V-  Sz'  +  2(gl  -  d], 
a 

=  -[-  (bz  +  d)  ±2  VZ22  -  2{A- lEJz  +  A-r  -  2iB], 


then 


1 


w  =  -  [-  (62  +  5)  ±  2  VZ  V2'  -  2002  +  (A, 


36        COMPLEX   CONICS  AND  THEIK   REAL  REPRESENTATION. 

and 

iv  =  -[-  (62  +  6)  ±  vi-  n     z'  =  ^.,     z  =  o. 

a.  z\X 

If  Z  =1=  0,  we  have  for  the  roots  of  the  radical 

z',  2     =  2  ±  2 =  20  ±  a'^'/,  /  =  I V2  ^  3. 

The  corresponding  values  of  w  are 

IV  ,  w     =  ^  T 7^ =  Wr)  =F  ba  '■''I. 

If  the  conic  is  real,  it  is  known^  that  the  points  corresponding 
to  the  roots  of  the  radical  are  the  two  real  foci  of  the  conic  in 
the  case  Z  4=  0,  and  the  real  finite  focus  in  the  case  Z  =  0. 
In  the  case  of  imaginary  conies  then  z'  -^  10'  and  z"  — »  10"  are 
the  pictures  of  the  foci,  now  imaginar}',  corresponding  to  the 
real  foci  in  the  real  case. 

Equation  (!')  is  of  the  second  degree  in  either  of  the  variables, 
hence  a  two-sheeted  Reimann  surface  is  required  for  complete 
depiction  in  case  either  is  expressed  as  a  function  of  the  other. 
The  obvious  disadvantage  of  proceeding  in  this  way  is  that  an 
irrationality  is  introduced.  We  avoid  this  by  introducing  a 
parameter  as  in  the  real  case.  This  we  effect  in  the  following 
manner : 

The  slope  of  (1)  is  given  by 

dt]  a^  -\-  ^Tf]  -\-  d 

dk  "  ~  ^^-\-cv  -\-~e' 
The  points  where  the  tangent  is  perpendicular  to  the  ^-axis  is 
given  by  intersection  of   (1)   and  ^^  +  crj  +  e  =  0.     The  ab- 
scissas of  these  points  are  the  roots  of 

(6)  Ze  -  2A^  4-  A  =  0 

or 

(0  _   ^  

7  =  Ve  -^  Z  =  V-  2  4-  3. 

The  corresponding  ordinates  are 

(8)  VI,  V2  =  r7oT^j8c-l'2/. 

1  Goursat,  Nouvclles  Annalles  Mathematique  (1887).  Cesaro,  Nouvelles 
Annalles  Mathematique  (1901). 


THE  REAL  REPRESENTATION  OF  THE  COMPLEX  CONIC. 

The  pictures  of  these  points  are 


37 


i^uVi),        Zi'-^wi] 


Z\,  Z2  =  Zo  ±  I  — Yir~   -'j 


Wi,  W2   =   Wo  ±  Z 


a/2 


Forming  the  pencil  of  lines  through  (^1,  771)  we  have 

(9)  V  -  VI-  r{^-  ^1),         T  =s  +  it. 

With  the  variation  of  r  through  the  domain  of  complex  numbers 
the  movable  intersection  of  the  line  and  the  conic  describes  the 
conic.  Thus  we  introduce  the  parameter  and  expressing  ^,  77, 
tv  and  z  in  terms  of  it  we  have 

2^- 


(10) 


^  =  ^l- 


7;  =  771  - 


ce 


2V-c0r 


IV 


IV I  — 


Z  =    Z\   — 


2  V-ce(i+tV) 

c{a  +  2|8r  +  cr^) ' 

2^l-cQ{\-ir) 
cia^-  2i3r  +  ct-)  ' 


c{a  +  2i3r  +  cr^) 

Thus  we  have  expressed  iv  and  2  rationally  in  terms  of  a  para- 
meter and  we  proceed  at  once  to  a  detailed  study  of  the  functions 
thus  obtained.  We  shall  find  it  simpler  however  to  pass  through 
a  series  of  linear  transformations  of  the  parameter  and  thereby 
transform  the  functions  iv  and  z  into  forms  more  frequently  met 
with  in  function  theory.  To  this  end  we  notice  that  the  common 
denominator  of  the  above  expressions  may  be  wTitten  {ct  +  18)'' 
+  Z,  so  our  first  change  of  parameter  is  by  the  transformation 

t'    =CT+  ^, 

and  we  get 


^  =b 


V  =  Vi 


2a/-c9 

r'^  +  Z 


2<-cQ{t'  -13) 
c{t''  +  Z) 


IV  =  Wi  — 


2l    - 


2V-c9(c  -  i^  +  It') 
c{r''  +  Z) 

2  V^^(c  +  ^ig  -  ir') 
cir''  +  Z) 


Again  changing  the  parameter  by  the  transformation 


=  iVz 


1  -t" 
1  +  t" 


we  have  after  some  reductions 
ci/2/ /  1  \ 


38        COMPLEX  CONICS  AND  THEIR  REAL  REPRESENTATION. 


77  =  770  -  ^^-[(^  +  iAfZ)r"  +^-^], 


Finally: 

(11) 


w 


c  -  i^  +  Vz  „ 

"^1   =  7u72  '"    ' 


ic^ 


Zo  + 


T2 


+  ?'/3-Vz   ,, 


2C 


1/2 


W 


=  u'o  4-  /o-i,         cri  =  9  (  '^1  +  ~  ) ' 


Z   =  Zo  -\-  I(T2, 


For  a  detailed  study  of  the  Riemann  surfaces  belonging  to  w 
and  z  let  us  assemble  the  series  of  transformations  by  which  we 
have  changed  our  parameter  and  examine  each  turn.  We  have, 
first  of  all,  r  the  variable  slope  of  the  lines  of  the  pencil  through 

(^b  Vi), 

77  —  771   =  t(^   -  ^1),  T   =   5   +  it, 

whose  intersection  with  the  conic  gives  the  00  ^  complex  points 
of  the  conic. 

The  quantities  ^,  77,  w  and  z  expressed  in  terms  of  r  are 

2V-ce 


^  =  .^1  - 


V  =  Vi 


2V-cer 


2V   =   Wi  — 


2Al-ce(l+^>) 


Z   =   Z\ 


c{a  +  2)3r  +  cj-) 
Then  by  a  series  of  transformations, 

r'  =  CT-\-  ^, 

1  -r" 
1  +r'" 

c  -  i^  +  Vz   ,, 


2V-ce(l  -^V) 


=  ^VZ 


Tl    = 


IC 


1/2 


<^i 


0-2 


THE   REAL  REPRESENTATION   OF  THE  COMPLEX  CONIC.       39 

we  bring  the  functions  iv  and  z  to  the  final  form 

w  =  Wo  +  /o-i, 

S   =  So  +  1^2- 

We  examine  these  in  the  reverse  order  in  which  they  are  tabu- 
lated. Whatever  be  the  Riemann  surfaces  belonging  to  ai  and  a> 
equations  (11)  express  merely  integral  rational  transformations 
of  the  same.  The  first  consists  of  an  expansion  1 1  \  and  a  rota- 
tion Z  (/)  followed  by  a  translation  over  the  vector  wq.  In 
the  second  we  have  first  of  all  a  reflection  over  the  axis  of  reals 
of  0-2,  followed  by  transformations  identical  in  nature  with  those 
of  the  first  equation. 

In  §§  4  and  5  we  discussed  functions  very  similar  to  ci  and  a-y- 
There  the  constant  entering,  §  4,  (7)  and  §  5,  (3)  was  a  real 
number,  c^.  Here  we  have  the  complex  numbers  a  =  a  —  c 
—  2ij3  and  y  =  a  —  c  -\-  2i/3  appearing  in  the  place  of  c^.  Let  us 
examine  the  Riemann  surface  belonging  to  ai.  For  the  moment 
let  us  write  o-i  =  u  +  iv,  n  =  x  +  iy  and  y  =  a  —  c  -\-  'lijS 
=  p  -\-  iq,  and  resolve  the  expression  into  its  real  and  pure 
imaginary  parts 


1/         qx  -  py\ 


We  seek  the  orthogonal  families  of  curves  in  the  cri-surface 
corresponding  to  concentric  circles  x~  -\-  y~  =  r^,  and  the  pencil 
of  rays  y  =  ex  in  the  ri-plane.     For  the  first  we  have: 

(^4  _  2pr^  +  R2)u2  _  4qrhii,  +  (r'  -f.  2pr'-  +  R'y  -  ^     ^^        =  0, 

where  R  =  Vp-  -{-  q^  =  \y\.  This  is  the  equation  of  a  conic 
for  which  the  discriminant,  (r^  —  R-)"  is  a  quantity  which  is 
never  less  than  zero;  hence  corresponding  to  the  family  of  con- 
centric circles  about  the  origin  in  the  ri-plane  is  a  family  of 
ellipses. 

The  curves  in  the  o-i-surface  corresponding  to  the  pencil  of 
rays  y  =  ex  in  the  ri-plane  are  given  by  the  equation, 

2c(ep  -  g)w-  +  2g(l  +  c')uv  -  2(p  -  eq)v-  -  [2cp 

-  q(l  -  c2)]V4(l  +  c')  =  0. 


40         COMPLEX  CONICS  AND  THEIR  REAL  REPRESENTATION. 

This  equation  represents  a  one-parameter  family  of  hyperbolas 
for  the  discriminant,  —  {2cp  —  q  +  qc^Y,  is  a  quantity  which 
is  never  greater  than  zero. 

We  find  further  that  the  real  foci  of  both  families  of  curves 
is  given  by  Vy  and  furthermore  the  slope  of  the  principal  axis 
is  given  by  {R  —  y)jq  and  this  is  the  tangent  of  Z  ( V7).  The 
(Ti-surface  then  consists  of  confocal  ellipses  and  hyperbolas 
corresponding  to  concentric  circles  about  and  radial  lines  through 
the  origin  of  the  ri-plane. 

Thus  we  see  that  the  Riemann  surface  belonging  to  the  func- 
tion ci  =  ^(n  +  (t/ti))  is  got  from  the  surface  belonging  to 
^(n  +  (l/ri))  by  a  magnification  in  the  ratio  1  :  |  V7I  and  a 
rotation  through  the  angle  of  V7,  Similar  results  hold  for  the 
function  <J2  using  a  instead  of  7. 

The  transformations 

c  -  r/3  +  VZ    „             ,                  c  +  2/3  -  VZ    „ 
Ti  = r-j7^ r  and         ti  =  r-r^ r 

transform  the  ry  and  T2-planes  into  themselves,  merely  inter- 
changing the  concentric  circles  among  themselves  and  the  pencil 
of  rays  among  themselves. 

Thus  as  t"  describes  a  circle  about  the  origin  in  its  plane  ri 
and  T2  describe  circles  about  the  origins  in  their  respective  planes 
and  in  turn  ai  and  <X2  describe  ellipses  in  their  respective  planes 
with  centers  at  the  origin;  while  finally  iv  and  z  describe  ellipses 
about  M'o  and  20  in  their  respective  planes.  Then  as  r"  describes 
a  ray  of  the  pencil  orthogonal  to  the  concentric  circles  w  and  2 
describe  hyperbolas  in  their  respective  planes  confocal  with  the 
ellipses  just  described. 

The  Riemann  surface  belonging  to  the  function 


""K^+O 


is  two-sheeted,  the  cross-cut  of  the  surface  extending  from  —  1 
to  +  1-  The  unit  circle  of  the  r-plane  maps  into  this  line  con- 
sidered as  double.  Circles  outside  the  unit  circle  and  concentric 
with  it  map  into  ellipses  in  the  upper  sheet  of  the  o--surface. 
Circles  inside  the  unit  circle  and  concentric  with  it  map  into 
ellipses  lying  in  the  lower  sheet.     So  for  our  functions  cri  and  a-^. 


THE   REAL   REPRESENTATION   OF  THE   COMPLEX   CONIC.        41 


The  Riemann  surfaces  belonging  to  them  are  two-sheeted,  the 
sheets  being  joined  along  the  lines  joining  —  V7  and  +  V7  or 
—  Va  and  +  Va  as  the  case  may  be.  To  the  circle  of  radius 
I  V7 1  of  the  n-plane  corresponds  this  line  joining  the  branch 
points  ±  V7  counted  twice.  Ellipses  in  the  ci-surface,  corre- 
sponding to  circles  lying  outside  the  circle  of  radius  \  ^y\  in 
the  n-plane  lie  in  the  upper  or  first  sheet  of  the  surface,  while 
those  corresponding  to  circles  within  the  circle  of  radius  |  V7I 
lie  in  the  lower  or  second  sheet.     As  we  have  already  noted, 


iiri 


>C 


/ 


Fig.  3. 


42         COMPLEX  CONICS  AND  THEIR  REAL  REPRESENTATION. 

the  IV-  and  z-surfaces  are  integral  rational  transformations  of 
the  ci-  and  <72-surfaces  respectively. 

With  proper  precautions  we  may  consider  all  these  surfaces 
superposed  and  with  origins  common  with  that  of  the  (^,  77)- 
plane.  The  surfaces  discussed  are  to  be  considered  in  no  wise 
organically  connected  with  each  other.  A  schematic  diagram  of 
quantities  considered  is  set  forth  in  drawings. 

The  Case  Z  =  0. 

17.  So  long  as  we  require  that  the  complex  coefficient  jS  in  our 
so-called  canonical  equation  of  the  conic  be  a  general  complex 
quantity,  that  is  /3  =  61  +  ib2  with  61  and  bo  different  from  zero, 
this  case,  Z  =  0  cannot  occur;  for  we  must  have 

Z  =  ac  -  i82  =  ac  -  bi"  +  62'  -  216162  =  0, 

which  requires  that  either  61  =  0  or  62  =  0. 

This  case  requires,  then,  that  either  61  or  62  vanish.  If  62  =  0, 
the  conic  becomes  real  and  we  have  already  considered  this 
case.  Chapter  I.  Again  Z  may  vanish  for  61  =  0,  that  is,  for 
jS  =  ib2,  pure  imaginary.     For  this  case  we  have 

Z  =  ac  -  /3-  =  ac  +  62-  =  0. 

Hence  if  |acl  =  62"  and  a  and  c  have  opposite  signs,  Z  will  be 
zero.  Since  bo  =  ±i '^ac,  the  terms  of  second  degree  for  this 
case,  just  as  for  the  real  case,  form  a  perfect  square: 

a^2  _t-  2ib>i^r}  -  erf  =  ( Va^  ±  ^  <cr}f. 

Choosing  the  plus  sign  we  have  for  the  equation  of  the  conic 
with  Z  =  0: 

a^2  -f  2i  Vac^?7  -  erf-  +  2d^  +  2er)  +  T  =  0. 

The  equation  in  z  and  w  giving  the  real  representation  is 
aV  +  2b'wz  +  c'z"  +  28io  +  2€z  +  i'  =  0, 


where 


a'  =  a  +  c  -^2^ae  =  i-^-}-  ^ef, 
c'  =  a  +  c  —  2  ^  =  ( Va  —  Vc)2, 
h'  =  a  —  c, 


THE  REAL  REPRESENTATION  OF  THE  COMPLEX  CONIC.   43 

8  =  2{d-  ie), 

e  =  2{d  +  ie)  =  5, 

We  resort  to  the  use  of  the  parameter  in  this  case  as  in  the 
preceding,  with  the  same  method  of  introducing  it. 

The  quadratic  giving  the  points  where  the  tangent  is  vertical 
reduces  for  this  case  to  —  2A^  +  A  =  0,  which  with 

i3^  -  C77  +  e  =  0 
gives  for  the  coordinates  of  the  point 

A 


^1  = 


2A* 


irr     vei 


Introducing  the  parameter  as  in  the  preceding  section  we  have 

77  -  Tji  =  r(^  -  ^i),         T  =  s  -\-  it, 

which  considered  with  (1)  gives  for  ^,  77,  w  and  z  expressed  as 
functions  of  r 


^  =  ^1  + 
^  =  m  + 


2A 


{cr  -  ^f 
2Ar 


=  ^1  + 


=  T/l  -}- 


2a/^ 


2 -Veer 


Making  the  transformations 


T    = 


cr  -  /3  =  r', 


w  =  wi  -\- 

2  =  Sl  + 

2(c  -  il3) 
Ti  —  i 

2(0  -  i^) 

T2   —  i 


2A(1  +  Jt) 
(cr  -  ^f 

2A(1  +  ir) 


of  the  parameter  and  applying  them  respectively  to  the  func- 


44         COMPLEX   CONICS  AND  THEIR  REAL  REPRESENTATION. 

tions  20  and  z  we  have  after  reduction 

A  Ve 

A  ^f 

where  6  =  ( Vc(^  +  i  Vae)^  is  the  determinant  of  the  conic, 
z'  ->  w'  is  the  picture  of  the  finite  focus.  The  functions  <ti  and  era 
are  well  known  and  have  been  discussed  in  Chapter  I.  The 
functions  w  and  z  are  linear  integral  transformations  of  the 
surfaces  belonging  to  a-i  and  0-2. 

18.  The  Case,  h  =  0. — In  §  13  we  noted  that  in  the  case 
b  =  a  -\-  c  =  0  our  variables  iv  and  z  were  rationally  separable. 
We  have 

mv^  +  28iv  =  -  yz-  -  28z  -  4^, 

Let  us  put 
Then 

,2  2  -/2 

W       =   p-^  —   2    . 

This  case  is  similar  to  the  one  discussed  by  Holzmiiller, 

?(J  =  Vl  —  s^. 

The    complex    quantity   p    however   excludes   the    involutorial 
property  of  the  case  just  cited.     The  simplification  of  this  case 
is  noted  by  observing  equations  (9)  of  section  (16). 
Jackson,  Mississippi. 


VITA. 

Benjamin  Ernest  Mitchell,  son  of  B.  W.  and  Rachel  Mitchell, 
born  Oct.  16,  1879,  Morrisville,  Mo.  Graduated,  Scarritt-Mor- 
risville  College,  A.B.,  1900.  Student,  University  of  Missouri, 
1902-03.  Professor  of  Mathematics,  Scarritt-Morrisville  Col- 
lege, 1903-06.  Scholastic  Fellow,  1906-07;  Teaching  Fellow, 
1907-08;  A.M.,  1908;  Instructor  in  Mathematics,  1908-12;  Van- 
derbilt  University.  Student,  Columbia  University,  1912-14. 
Tutor  in  Mathematics,  College  of  the  City  of  New  York,  1913. 
Instructor  in  Extension  Teaching,  Columbia  University,  1913-14. 
Professor  of  Mathematics,  Millsaps  College,  1914-.  Member 
of  American  Mathematical  Society.  My  sincere  thanks  to  the 
members  of  the  Department  of  Mathematics  of  Columbia  Uni- 
versity and  in  a  special  degree  to  Professors  Kasner  and  Keyser. 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  50  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.00  ON  THE  SEVENTH  DAY 
OVERDUE. 


APH    22  ]£35 

MAR  Z  0  1979 

1 

Btc.  uiK.  ijAR  1  5  197 

) 

LD  21-100m-8,'34 

UNIVERSITY  OF  CALIFORNIA  LIBRARY 


U    C    BERKELEY  LIBRARIES 

111111  |l|||l|i||| 


CDblBSlflMb 


